Clustering and classification

date()

## [1] "Sat Nov 26 09:14:30 2022"

Short description of dataset

library(MASS)
# load the data 
data("Boston")

# check structure and dimensions
str(Boston)

## 'data.frame':    506 obs. of  14 variables:
##  $ crim   : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
##  $ zn     : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
##  $ indus  : num  2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
##  $ chas   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ nox    : num  0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
##  $ rm     : num  6.58 6.42 7.18 7 7.15 ...
##  $ age    : num  65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
##  $ dis    : num  4.09 4.97 4.97 6.06 6.06 ...
##  $ rad    : int  1 2 2 3 3 3 5 5 5 5 ...
##  $ tax    : num  296 242 242 222 222 222 311 311 311 311 ...
##  $ ptratio: num  15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
##  $ black  : num  397 397 393 395 397 ...
##  $ lstat  : num  4.98 9.14 4.03 2.94 5.33 ...
##  $ medv   : num  24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...

dim(Boston)

## [1] 506  14

The dataset contains housing values in suburbs of Boston. It can be downloaded from R’s MASS package, and contains 506 observations of 14 variables. More information on the data and variables can be found here: https://stat.ethz.ch/R-manual/R-devel/library/MASS/html/Boston.html

Graphical overview of the data and summaries of the variables

library(tidyr)
# check summary of data
summary(Boston)

##       crim                zn             indus            chas        
##  Min.   : 0.00632   Min.   :  0.00   Min.   : 0.46   Min.   :0.00000  
##  1st Qu.: 0.08205   1st Qu.:  0.00   1st Qu.: 5.19   1st Qu.:0.00000  
##  Median : 0.25651   Median :  0.00   Median : 9.69   Median :0.00000  
##  Mean   : 3.61352   Mean   : 11.36   Mean   :11.14   Mean   :0.06917  
##  3rd Qu.: 3.67708   3rd Qu.: 12.50   3rd Qu.:18.10   3rd Qu.:0.00000  
##  Max.   :88.97620   Max.   :100.00   Max.   :27.74   Max.   :1.00000  
##       nox               rm             age              dis        
##  Min.   :0.3850   Min.   :3.561   Min.   :  2.90   Min.   : 1.130  
##  1st Qu.:0.4490   1st Qu.:5.886   1st Qu.: 45.02   1st Qu.: 2.100  
##  Median :0.5380   Median :6.208   Median : 77.50   Median : 3.207  
##  Mean   :0.5547   Mean   :6.285   Mean   : 68.57   Mean   : 3.795  
##  3rd Qu.:0.6240   3rd Qu.:6.623   3rd Qu.: 94.08   3rd Qu.: 5.188  
##  Max.   :0.8710   Max.   :8.780   Max.   :100.00   Max.   :12.127  
##       rad              tax           ptratio          black       
##  Min.   : 1.000   Min.   :187.0   Min.   :12.60   Min.   :  0.32  
##  1st Qu.: 4.000   1st Qu.:279.0   1st Qu.:17.40   1st Qu.:375.38  
##  Median : 5.000   Median :330.0   Median :19.05   Median :391.44  
##  Mean   : 9.549   Mean   :408.2   Mean   :18.46   Mean   :356.67  
##  3rd Qu.:24.000   3rd Qu.:666.0   3rd Qu.:20.20   3rd Qu.:396.23  
##  Max.   :24.000   Max.   :711.0   Max.   :22.00   Max.   :396.90  
##      lstat            medv      
##  Min.   : 1.73   Min.   : 5.00  
##  1st Qu.: 6.95   1st Qu.:17.02  
##  Median :11.36   Median :21.20  
##  Mean   :12.65   Mean   :22.53  
##  3rd Qu.:16.95   3rd Qu.:25.00  
##  Max.   :37.97   Max.   :50.00

# draw pairs plot of the variables
pairs(Boston)

# calculate the correlation matrix and round it
cor_matrix <- cor(Boston)
cor_matrix %>% round(2)

##          crim    zn indus  chas   nox    rm   age   dis   rad   tax ptratio
## crim     1.00 -0.20  0.41 -0.06  0.42 -0.22  0.35 -0.38  0.63  0.58    0.29
## zn      -0.20  1.00 -0.53 -0.04 -0.52  0.31 -0.57  0.66 -0.31 -0.31   -0.39
## indus    0.41 -0.53  1.00  0.06  0.76 -0.39  0.64 -0.71  0.60  0.72    0.38
## chas    -0.06 -0.04  0.06  1.00  0.09  0.09  0.09 -0.10 -0.01 -0.04   -0.12
## nox      0.42 -0.52  0.76  0.09  1.00 -0.30  0.73 -0.77  0.61  0.67    0.19
## rm      -0.22  0.31 -0.39  0.09 -0.30  1.00 -0.24  0.21 -0.21 -0.29   -0.36
## age      0.35 -0.57  0.64  0.09  0.73 -0.24  1.00 -0.75  0.46  0.51    0.26
## dis     -0.38  0.66 -0.71 -0.10 -0.77  0.21 -0.75  1.00 -0.49 -0.53   -0.23
## rad      0.63 -0.31  0.60 -0.01  0.61 -0.21  0.46 -0.49  1.00  0.91    0.46
## tax      0.58 -0.31  0.72 -0.04  0.67 -0.29  0.51 -0.53  0.91  1.00    0.46
## ptratio  0.29 -0.39  0.38 -0.12  0.19 -0.36  0.26 -0.23  0.46  0.46    1.00
## black   -0.39  0.18 -0.36  0.05 -0.38  0.13 -0.27  0.29 -0.44 -0.44   -0.18
## lstat    0.46 -0.41  0.60 -0.05  0.59 -0.61  0.60 -0.50  0.49  0.54    0.37
## medv    -0.39  0.36 -0.48  0.18 -0.43  0.70 -0.38  0.25 -0.38 -0.47   -0.51
##         black lstat  medv
## crim    -0.39  0.46 -0.39
## zn       0.18 -0.41  0.36
## indus   -0.36  0.60 -0.48
## chas     0.05 -0.05  0.18
## nox     -0.38  0.59 -0.43
## rm       0.13 -0.61  0.70
## age     -0.27  0.60 -0.38
## dis      0.29 -0.50  0.25
## rad     -0.44  0.49 -0.38
## tax     -0.44  0.54 -0.47
## ptratio -0.18  0.37 -0.51
## black    1.00 -0.37  0.33
## lstat   -0.37  1.00 -0.74
## medv     0.33 -0.74  1.00

# visualize the correlation matrix
library(corrplot)

## corrplot 0.92 loaded

corrplot(cor_matrix, method="circle")

The summary of the data shows means, medians and ranges of the different variables, for example median number of rooms per dwelling is 6.2 (range 3.6-8.8).

Tax rate and accessibility to radial highways seem to be strongly positively correlated. Median value and lower status of the population are negatively correlated. The plotted correlation matrix shows also other positive and negative correlations. The darker blue a sphere is, the stronger the positive correlation between variables, and the darker red a sphere is, the stronger the negative correlation.

Standardizing the dataset

library(dplyr)

## 
## Attaching package: 'dplyr'

## The following object is masked from 'package:MASS':
## 
##     select

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

# center and standardize variables
boston_scaled <- scale(Boston)

# summaries of the scaled variables
summary(boston_scaled)

##       crim                 zn               indus              chas        
##  Min.   :-0.419367   Min.   :-0.48724   Min.   :-1.5563   Min.   :-0.2723  
##  1st Qu.:-0.410563   1st Qu.:-0.48724   1st Qu.:-0.8668   1st Qu.:-0.2723  
##  Median :-0.390280   Median :-0.48724   Median :-0.2109   Median :-0.2723  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.007389   3rd Qu.: 0.04872   3rd Qu.: 1.0150   3rd Qu.:-0.2723  
##  Max.   : 9.924110   Max.   : 3.80047   Max.   : 2.4202   Max.   : 3.6648  
##       nox                rm               age               dis         
##  Min.   :-1.4644   Min.   :-3.8764   Min.   :-2.3331   Min.   :-1.2658  
##  1st Qu.:-0.9121   1st Qu.:-0.5681   1st Qu.:-0.8366   1st Qu.:-0.8049  
##  Median :-0.1441   Median :-0.1084   Median : 0.3171   Median :-0.2790  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.5981   3rd Qu.: 0.4823   3rd Qu.: 0.9059   3rd Qu.: 0.6617  
##  Max.   : 2.7296   Max.   : 3.5515   Max.   : 1.1164   Max.   : 3.9566  
##       rad               tax             ptratio            black        
##  Min.   :-0.9819   Min.   :-1.3127   Min.   :-2.7047   Min.   :-3.9033  
##  1st Qu.:-0.6373   1st Qu.:-0.7668   1st Qu.:-0.4876   1st Qu.: 0.2049  
##  Median :-0.5225   Median :-0.4642   Median : 0.2746   Median : 0.3808  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 1.6596   3rd Qu.: 1.5294   3rd Qu.: 0.8058   3rd Qu.: 0.4332  
##  Max.   : 1.6596   Max.   : 1.7964   Max.   : 1.6372   Max.   : 0.4406  
##      lstat              medv        
##  Min.   :-1.5296   Min.   :-1.9063  
##  1st Qu.:-0.7986   1st Qu.:-0.5989  
##  Median :-0.1811   Median :-0.1449  
##  Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.6024   3rd Qu.: 0.2683  
##  Max.   : 3.5453   Max.   : 2.9865

# change the object to data frame
boston_scaled <- as.data.frame(boston_scaled)

boston_scaled$crim <- as.numeric(boston_scaled$crim)

# create a quantile vector of crim
bins <- quantile(boston_scaled$crim)

# create a categorical variable 'crime'
crime <- cut(boston_scaled$crim, breaks = bins, label = c("low", "med_low", "med_high", "high"), include.lowest = TRUE)

# remove original crim from the dataset
boston_scaled <- dplyr::select(boston_scaled, -crim)

# add the new categorical value to scaled data
boston_scaled <- data.frame(boston_scaled, crime)

# number of rows in the Boston dataset 
n <- nrow(boston_scaled)

# choose randomly 80% of the rows
ind <- sample(n,  size = n * 0.8)

# create train set
train <- boston_scaled[ind,]

# create test set 
test <- boston_scaled[-ind,]

The variables are now transformed on the same scale, so now we can compare them.

Linear discriminant analysis and predictions

# linear discriminant analysis
lda.fit <- lda(crime ~., data = train)

# print the lda.fit object
lda.fit

## Call:
## lda(crime ~ ., data = train)
## 
## Prior probabilities of groups:
##       low   med_low  med_high      high 
## 0.2326733 0.2549505 0.2500000 0.2623762 
## 
## Group means:
##                  zn      indus         chas        nox          rm        age
## low       0.8939510 -0.9754278 -0.104792887 -0.8485044  0.43405146 -0.8736005
## med_low  -0.1246578 -0.2867301 -0.004759149 -0.5745673 -0.10386054 -0.3349787
## med_high -0.3929954  0.2851589  0.234426408  0.4297635  0.09979558  0.4504816
## high     -0.4872402  1.0170298 -0.086616792  1.0343800 -0.41806290  0.7834562
##                 dis        rad        tax     ptratio       black        lstat
## low       0.8222879 -0.6825367 -0.7269244 -0.45365083  0.37315387 -0.750377064
## med_low   0.3216926 -0.5592799 -0.4881197 -0.09336749  0.32650788 -0.133788700
## med_high -0.4346438 -0.4360651 -0.2968440 -0.31971579  0.08686171  0.007848367
## high     -0.8364898  1.6390172  1.5146914  0.78181164 -0.76110497  0.840072897
##                 medv
## low       0.52446690
## med_low   0.01469537
## med_high  0.20108198
## high     -0.67738454
## 
## Coefficients of linear discriminants:
##                  LD1         LD2         LD3
## zn       0.144675133  0.62018262 -0.96325232
## indus    0.150676733 -0.60868188  0.42780791
## chas    -0.042350476 -0.02561201  0.04412303
## nox      0.372367898 -0.53473272 -1.44290314
## rm       0.008221839 -0.08137991 -0.09785932
## age      0.238992718 -0.30502986 -0.07271449
## dis     -0.029520974 -0.16386419  0.18039555
## rad      3.478432985  0.78820916  0.10942887
## tax     -0.091702733  0.18587134  0.35045796
## ptratio  0.125339980  0.05651590 -0.35584810
## black   -0.096920528  0.03746421  0.13243900
## lstat    0.148232788 -0.15750481  0.32515880
## medv     0.053729282 -0.34615529 -0.27116387
## 
## Proportion of trace:
##    LD1    LD2    LD3 
## 0.9511 0.0369 0.0120

# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
  heads <- coef(x)
  arrows(x0 = 0, y0 = 0, 
         x1 = myscale * heads[,choices[1]], 
         y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
  text(myscale * heads[,choices], labels = row.names(heads), 
       cex = tex, col=color, pos=3)
}

# target classes as numeric
classes <- as.numeric(train$crime)

# plot the lda results
plot(lda.fit, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit, myscale = 2)

# save the correct classes from test data
correct_classes <- test$crime

# remove the crime variable from test data
test <- dplyr::select(test, -crime)

# predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)

# cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class)

##           predicted
## correct    low med_low med_high high
##   low       21      11        1    0
##   med_low    3      16        4    0
##   med_high   0      11       12    2
##   high       0       0        0   21

It would seem that with high crime rate the predictions are most accurate. With low to medium and medium to high there is most inaccuracy.

Distance measures and k-means clustering

library(ggplot2)
set.seed(123)
data(Boston)
boston_scaled2 <- as.data.frame(scale(Boston))

# euclidean distance matrix
dist_eu <- dist(boston_scaled2)
summary(dist_eu)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.1343  3.4625  4.8241  4.9111  6.1863 14.3970

# k-means clustering
km <- kmeans(boston_scaled2, centers = 3)

# plot part of the Boston dataset with clusters
pairs(boston_scaled2[6:10], col = km$cluster)

# determine the maximum number of clusters
k_max <- 10

# calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(boston_scaled2, k)$tot.withinss})

# visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')

# k-means clustering
km <- kmeans(boston_scaled2, centers = 2)

# plot the Boston dataset with clusters
pairs(boston_scaled2, col = km$cluster)

# plot part of the Boston dataset with clusters
pairs(boston_scaled2[6:10], col = km$cluster)

From plotting the total within sum of squares we see that around two the value changes quite a lot, so the appropriate number of cluster would be two. The data seems to divide nicely between two clusters according to most variables.

Bonus

set.seed(123)
data(Boston)
boston_scaled3 <- as.data.frame(scale(Boston))

# k-means clustering
km2 <- kmeans(boston_scaled3, centers = 3)

# linear discriminant analysis
lda.fit2 <- lda(km2$cluster ~., data = boston_scaled3)

# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
  heads <- coef(x)
  arrows(x0 = 0, y0 = 0, 
         x1 = myscale * heads[,choices[1]], 
         y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
  text(myscale * heads[,choices], labels = row.names(heads), 
       cex = tex, col=color, pos=3)
}

# target classes as numeric
classes <- as.numeric(km2$cluster)

# plot the lda results
plot(lda.fit2, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit2, myscale = 4)

Variables age, black, and tax seem to be the most influential linear separators for the clusters.

Super-Bonus

library(plotly)

## 
## Attaching package: 'plotly'

## The following object is masked from 'package:ggplot2':
## 
##     last_plot

## The following object is masked from 'package:MASS':
## 
##     select

## The following object is masked from 'package:stats':
## 
##     filter

## The following object is masked from 'package:graphics':
## 
##     layout

lda.fit3 <- lda(crime ~., data = train)
model_predictors <- dplyr::select(train, -crime)

# check the dimensions
dim(model_predictors)

## [1] 404  13

dim(lda.fit3$scaling)

## [1] 13  3

# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda.fit3$scaling
matrix_product <- as.data.frame(matrix_product)

classes <- as.numeric(train$crime)
train2 <- dplyr::select(train, -crime)
km3 <- kmeans(train2, centers = 4)
clusters <- as.numeric(km3$cluster)

plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color=classes)

plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color=clusters)

The cluster that is on the right-hand side (at approximately x=-4 and y=2) looks quite same in both plots. In the first plot the clusters are more defined, in the second one they mix more with each other.